Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 1998 • Question 2

Question

Math 6420 • PDE Fall 1998 • Question 2

Consider the integral equation \[ u(x)=f(x)+\frac {1}{2}\int _0^1 xtu(t)\,dt,\qquad \text {for }0\le x\le 1 \] where \(f(x)\) is a prescribed function.

  1. Show that the equation has at most one solution.
  2. Derive a formula for the solution.
  3. Calculate the solution for the special case \(f(x)=5x/6\).

Raw LaTeX

Consider the integral equation
\[
  u(x)=f(x)+\frac{1}{2}\int_0^1 xtu(t)\,dt,\qquad \text{for }0\le x\le 1
\]
where \(f(x)\) is a prescribed function.
\begin{enumerate}
\item[a).] Show that the equation has at most one solution.
\item[b).] Derive a formula for the solution.
\item[c).] Calculate the solution for the special case \(f(x)=5x/6\).
\end{enumerate}

Topic Tags

Cauchy Integral Formula and Integral Theorems Distributions and Weak Solutions Entropy Solutions, Shocks, and Rarefactions Linear Programming and Optimization